MATH

the fist term of a geometric series is 1, the nth term is 128 and the sum of the n term is 225. Find the common ratio and the number of terms?

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  1. If the sum of the n term = 225

    your question does not make sense.

    It can not be solved.

    But if the sum of the n term = 255 then :

    In geometric sequence :

    The nth term is :

    an = a1 * r ^ ( n - 1 )

    Where a1 is the first term of the sequence.

    r is the common ratio.

    n is the number of the terms

    The sum of the first n terms is given by:

    S = a1 * [ ( 1 - r ^ n ) / ( 1 - r ) ]

    In this case :

    a1 = 1

    an = a1 * r ^ ( n - 1 ) = 1 * r ^ ( n - 1 ) = r ^ ( n - 1 ) = 128

    r ^ ( n - 1 ) = 128

    S = a1 * [ ( 1 - r ^ n ) / ( 1 - r ) ] = 1 * [ ( 1 - r ^ n ) / ( 1 - r ) ] = ( 1 - r ^ n ) / ( 1 - r ) = 255

    ( 1 - r ^ n ) / ( 1 - r ) = 255

    So :

    r ^ ( n - 1 ) = 128 Multiplye both sides by r

    r ^ ( n - 1 ) * r = 128 r

    r ^ n = 128 r

    Becouse r ^ ( n - 1 ) * r = r ^ n

    Now :

    r ^ n = 128 r

    You already know :

    ( 1 - r ^ n ) / ( 1 - r ) = 255

    ( 1 - 128 r ) / ( 1 - r ) = 255 Multiply both sides by ( 1 - r )

    1 - 128 r = 255 ( 1 - r )

    1 - 128 r = 255 - 255 r Subtract 1 to both sides

    1 - 128 r - 1 = 255 - 255 r - 1

    - 128 r = 254 - 255 r Add 255 r to both sides

    - 128 r + 255 r = 254

    127 r = 254 Divide both sides by 127

    r = 254 / 127

    r = 2

    Also you already know :

    r ^ ( n - 1 ) = 128

    In this case :

    2 ^ ( n - 1 ) = 128 Take the logarithm of both sides

    ( n - 1 ) * log ( 2 ) = log ( 128 ) Divide both sides by log ( 2 )

    n - 1 = log ( 128 ) / log ( 2 )

    n - 1 = 7 Add 1 to both sides

    n - 1 + 1 = 7 + 1

    n = 8

    ________________________________________

    Remark:

    log [ 2 ^ ( n - 1 ) ] = ( n - 1 ) * log ( 2 ) becouse

    log ( a ^ x ) = x * log ( a )

    In this case :

    a = 2 , x = n - 1
    ________________________________________

    Solutions :

    Common ratio

    r = 2

    Number of the terms

    n = 8

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  2. I agree with Bosnian that the question contains a typo.
    If the sum of n terms is 225 as stated we could solve for r = 2.309..
    but then the solution for n is not a whole number.

    Bosnian assumed correctly that sum(8) = 255

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